Only 18% of likely voters have a favorable impression of Greene. DeMint, by contrast, has a 69% favorable rating.

Greene, who pledges to "reclaim our country" from "terrorists and the communists," hasn't had a good few weeks. Records from Greene's military service that called Greene a "threat to others" were revealed, and an L.A.-based consulting firm that had agreed to work with Greene parted ways with him after less than a week.

Only Rasmussen has polled the South Carolina Senate race so far. The pollster's first and only other survey, from June 10, showed DeMint leading Greene 58%-21%. The margin of error of the latest Rasmussen poll is Â±4.5 percentage points.